A036043 -id:A036043 - cp's OEIS frontend

A333485 Heinz numbers of all integer partitions sorted first by sum, then by decreasing length, and finally lexicographically. A code for the Fenner-Loizou tree A228100.

1, 2, 4, 3, 8, 6, 5, 16, 12, 9, 10, 7, 32, 24, 18, 20, 15, 14, 11, 64, 48, 36, 40, 27, 30, 28, 25, 21, 22, 13, 128, 96, 72, 80, 54, 60, 56, 45, 50, 42, 44, 35, 33, 26, 17, 256, 192, 144, 160, 108, 120, 112, 81, 90, 100, 84, 88, 75, 63, 70, 66, 52, 49, 55, 39, 34, 19

Offset: 0

Gus Wiseman, May 11 2020

A permutation of the positive integers.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), which gives a bijective correspondence between positive integers and integer partitions.
As a triangle with row lengths A000041, the sequence starts {{1},{2},{4,3},{8,6,5},...}, so offset is 0.

			The sequence of terms together with their prime indices begins:
    1: {}              11: {5}                 56: {1,1,1,4}
    2: {1}             64: {1,1,1,1,1,1}       45: {2,2,3}
    4: {1,1}           48: {1,1,1,1,2}         50: {1,3,3}
    3: {2}             36: {1,1,2,2}           42: {1,2,4}
    8: {1,1,1}         40: {1,1,1,3}           44: {1,1,5}
    6: {1,2}           27: {2,2,2}             35: {3,4}
    5: {3}             30: {1,2,3}             33: {2,5}
   16: {1,1,1,1}       28: {1,1,4}             26: {1,6}
   12: {1,1,2}         25: {3,3}               17: {7}
    9: {2,2}           21: {2,4}              256: {1,1,1,1,1,1,1,1}
   10: {1,3}           22: {1,5}              192: {1,1,1,1,1,1,2}
    7: {4}             13: {6}                144: {1,1,1,1,2,2}
   32: {1,1,1,1,1}    128: {1,1,1,1,1,1,1}    160: {1,1,1,1,1,3}
   24: {1,1,1,2}       96: {1,1,1,1,1,2}      108: {1,1,2,2,2}
   18: {1,2,2}         72: {1,1,1,2,2}        120: {1,1,1,2,3}
   20: {1,1,3}         80: {1,1,1,1,3}        112: {1,1,1,1,4}
   15: {2,3}           54: {1,2,2,2}           81: {2,2,2,2}
   14: {1,4}           60: {1,1,2,3}           90: {1,2,2,3}
The triangle begins:
    1
    2
    4   3
    8   6   5
   16  12   9  10   7
   32  24  18  20  15  14  11
   64  48  36  40  27  30  28  25  21  22  13
  128  96  72  80  54  60  56  45  50  42  44  35  33  26  17

Michael De Vlieger, Table of n, a(n) for n = 0..9295 (rows 0 <= n <= 25, flattened)
Michael De Vlieger, log-log plot of rows 0 <= n <= 30 of this sequence, highlighting 2^n in red and prime(n) in blue.
T. I. Fenner, G. Loizou: A binary tree representation and related algorithms for generating integer partitions. The Computer J. 23(4), 332-337 (1980)
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
The constructive version is A228100.
Sorting by increasing length gives A334433.
The version with rows reversed is A334438.
Sum of prime indices is A056239.
Reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Lexicographically ordered partitions are A193073.
Graded Heinz numbers are A215366.
Sorting partitions by Heinz number gives A296150.
If the fine ordering is by Heinz number instead of lexicographic we get A333484.
Cf. A000041, A026791, A036036, A036037, A036043, A185974, A211992, A228351, A296773, A333483, A334301, A334439.

ralensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]>Length[c],OrderedQ[{f,c}]];
Join@@Table[Times@@Prime/@#&/@Sort[IntegerPartitions[n],ralensort],{n,0,8}]

A001221(a(n)) = A115623(n).
A001222(a(n - 1)) = A331581(n).
A061395(a(n > 1)) = A128628(n).

Name extended by Peter Luschny, Dec 23 2020

A344089 Flattened tetrangle of reversed strict integer partitions, sorted first by length and then colexicographically.

Original entry on oeis.org

1, 2, 3, 1, 2, 4, 1, 3, 5, 2, 3, 1, 4, 6, 2, 4, 1, 5, 1, 2, 3, 7, 3, 4, 2, 5, 1, 6, 1, 2, 4, 8, 3, 5, 2, 6, 1, 7, 1, 3, 4, 1, 2, 5, 9, 4, 5, 3, 6, 2, 7, 1, 8, 2, 3, 4, 1, 3, 5, 1, 2, 6, 10, 4, 6, 3, 7, 2, 8, 1, 9, 2, 3, 5, 1, 4, 5, 1, 3, 6, 1, 2, 7, 1, 2, 3, 4

Offset: 0

Gus Wiseman, May 12 2021

First differs from the revlex (instead of colex) version for partitions of 12.
The zeroth row contains only the empty partition.
A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (3)(12)
  4: (4)(13)
  5: (5)(23)(14)
  6: (6)(24)(15)(123)
  7: (7)(34)(25)(16)(124)
  8: (8)(35)(26)(17)(134)(125)
  9: (9)(45)(36)(27)(18)(234)(135)(126)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724 plus one.
Taking lex instead of colex gives A026793 (non-reversed: A118457).
Triangle sums are A066189.
Reversing all partitions gives A344090.
The non-strict version is A344091.
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
Cf. A005117, A014466, A209862, A325683, A325859.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A246688, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344087, A344088, A344089.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

Table[Reverse/@Sort[Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,30}]

A111786 Array used to obtain the complete symmetric function in n variables in terms of the elementary symmetric functions; irregular triangle T(n,k), read by rows, with n >= 1 and 1 <= k <= A000041(n).

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -1, 2, 1, -3, 1, 1, -2, -2, 3, 3, -4, 1, -1, 2, 2, 1, -3, -6, -1, 4, 6, -5, 1, 1, -2, -2, -2, 3, 6, 3, 3, -4, -12, -4, 5, 10, -6, 1, -1, 2, 2, 2, 1, -3, -6, -6, -3, -3, 4, 12, 6, 12, 1, -5, -20, -10, 6, 15, -7, 1, 1, -2, -2, -2, -2, 3, 6, 6, 3, 3, 6, 1, -4, -12, -12, -12, -12, -4, 5, 20, 10, 30, 5, -6, -30, -20, 7, 21, -8, 1, -1

Offset: 1

Wolfdieter Lang, Aug 23 2005

The unsigned numbers give A048996. They are not listed on pp. 831-832 of Abramowitz and Stegun (reference given in A103921). One could call these numbers M_0 (like M_1, M_2, M_3 given in A036038, A036039, A036040, resp.).
The sequence of row lengths is A000041(n) (partition numbers).
The sign is (-1)^(n + m(n,k)) with m(n,k) the number of parts of the k-th partition of n taken in the mentioned order. For m(n,k), see A036043.
The row sum is 1 for n = 1, and 0 otherwise. The unsigned row sum is 2^(n-1) = A000079(n-1) for n >= 1.
The complete symmetric polynomial is also h(n; a[1],...,a[n]) = Det(A_n) with the matrix elements of the n X n matrix A_n given by A_n(k, k+1) = 1 for 1 <= k < n, A(k, m) = a[k-m+1] for n >=  k >= m >= 1, and 0 otherwise. [For an explanation of this statement, see the example for n = 4 below. See also p. 3 in MacMahon (1960).]

			Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
   1;
  -1,  1;
   1, -2,  1;
  -1,  2,  1, -3,  1;
   1, -2, -2,  3,  3, -4,  1;
  -1,  2,  2,  1, -3, -6, -1, 4, 6, -5, 1,
   ...
h(4; a[1],...,a[4])= -1*a[4] + 2*a[1]*a[3] + 1*a[2]^2 - 3*a[1]^2*a[2] + a[1]^4.
Consider variables x_1, x_2, x_3, x_4, and let a[1] = Sum_i x_i, a[2] = Sum_{i,j} x_i*x_j, a[3] = Sum_{i,j,k} x_i*x_j*x_k, and a[4] = x1*x2*x3*x4, where in all the sums no term is repeated twice.
Then h(4; a[1],...,a[4]) = Sum_i x_i^4 + Sum_{i,j} x_i^3*x_j + Sum_{i,j} x_i^2*x_j^2 + Sum_{i,j,k} x_i^2*x_j*x_k + Sum_{i,j,k,m} x_i*x_j*x_k*x_m, where again in all the sums no term is repeated twice. Thus, indeed, h is the complete symmetric polynomial in four variables x_1, x_2, x_3, x_4.

V. Krishnamurthy, Combinatorics, Ellis Horwood, Chichester, 1986, p. 55, eqs. (48) and (50).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Wolfdieter Lang, First 10 rows.
P. A. MacMahon, Combinatory analysis (2 vols.), Chelsea, NY, 1960; see p. 4.
OEIS, Orderings of partitions.

Cf. A000041, A000079, A036038, A036039, A036040, A036043, A048996, A103921, A105805, A115131, A210258.

The complete symmetric row polynomials h(n; a[1], ..., a[n]):= sum k over partitions of n of T(n, k)* A[k], with A[k] := a[1]^e(k, 1) * a[2]^e(k, 2) * ... * a[n]^e(k, n) is the k-th partition of n, in Abramowitz-Stegun order (see A105805 for this reference), is [1^e(k, 1), 2^e(k, 2), ..., n^e(k, n)], for k = 1..p(n), where p(n) = A000041(n) (partition numbers).
G.f.: A(x) = 1/(1 + Sum_{j = 1..infinity} (-1)^j * a[j]).
T(n, k) is the coefficient of x^n and a[1]^e(k, 1) * a[2]^e(k, 2) * ... * a[n]^e(k, n) in A(x) if the k-th partition of n, counted using the Abramowitz-Stegun order, is [1^e(k, 1), 2^e(k, 2), ..., n^e(k, n)] with e(k, j) >= 0 (and if e(k, j) = 0 then j^0 is not recorded).
T(n, k) = (-1)^(n + m(n, k)) * m(n, k)!/(Product_{j = 1..n} e(k, j)!), where m(n, k) := Sum_{j = 1..n} e(k, j), with [1^e(k, 1), 2^e(k, 2), ..., n^e(k, n)] being the k-th partition of n in the mentioned order. Here m(n, k) is the number of parts of the k-th partition of n. For m(n,k), see A036043.

Various sections edited by Petros Hadjicostas, Dec 15 2019

A335123 Minimum part of the n-th integer partition in Abramowitz-Stegun order (sum/length/lex); a(0) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 6, 3, 2, 1, 2, 1, 1, 1, 1, 1, 1, 7, 3, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 4, 3, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 4, 3, 2, 1, 3, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Gus Wiseman, May 24 2020

			Triangle begins:
  0
  1
  2 1
  3 1 1
  4 2 1 1 1
  5 2 1 1 1 1 1
  6 3 2 1 2 1 1 1 1 1 1
  7 3 2 1 2 1 1 1 1 1 1 1 1 1 1
  8 4 3 2 1 2 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1

OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
Partition minima of A334301.
The length of the same partition is A036043.
The Heinz number of the same partition is A334433.
The number of distinct parts in the same partition is A334440.
The maximum of the same partition is A334441.
The version for reversed partitions is A335124.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in (sum/length/revlex) order are A334439.
Cf. A026791, A049085, A103921, A124734, A185974, A193073, A334302, A334433.

Table[If[n==0,{0},Min/@Sort[IntegerPartitions[n]]],{n,0,8}]

a(n) = A055396(A334433(n)).

A335124 Minimum part of the n-th reversed integer partition in Abramowitz-Stegun order; a(0) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 1, 5, 1, 2, 1, 1, 1, 1, 6, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 7, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 8, 1, 2, 3, 4, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 9, 1, 2, 3, 4, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Gus Wiseman, May 24 2020

The ordering of reversed partitions is first by sum, then by length, and finally lexicographically. The version for non-reversed partitions is A335123.

			Triangle begins:
  0
  1
  2 1
  3 1 1
  4 1 2 1 1
  5 1 2 1 1 1 1
  6 1 2 3 1 1 2 1 1 1 1
  7 1 2 3 1 1 1 2 1 1 1 1 1 1 1
  8 1 2 3 4 1 1 1 2 2 1 1 1 1 2 1 1 1 1 1 1 1

OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
Partition minima of A036036.
The length of the same partition is A036043.
The maximum of the same partition is A049085.
The number of distinct parts in the same partition is A103921.
The Heinz number of the same partition is A185974.
The version for non-reversed partitions is A335123.
Lexicographically ordered reversed partitions are A026791.
Partitions in (sum/length/colex) order are A036037.
Partitions in opposite Abramowitz-Stegun (sum/length/revlex) order are A334439.
Cf. A124734, A193073, A334301, A334302, A334433, A334435.

Table[If[n==0,{0},Min/@Sort[Reverse/@IntegerPartitions[n]]],{n,0,8}]

a(n) = A055396(A185974(n)).

A344085 Triangle of squarefree numbers first grouped by greatest prime factor, then sorted by omega, then in increasing order, read by rows.

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 77, 66, 110, 154, 165, 231, 385, 330, 462, 770, 1155, 2310, 13, 26, 39, 65, 91, 143, 78, 130, 182, 195, 273, 286, 429, 455, 715, 1001, 390, 546, 858, 910, 1365, 1430, 2002, 2145, 3003, 5005, 2730, 4290, 6006, 10010, 15015, 30030

Offset: 1

Gus Wiseman, May 11 2021

Differs from A339195 in having 77 before 66.

			Triangle begins:
   1
   2
   3   6
   5  10  15  30
   7  14  21  35  42  70 105 210

Row lengths are A000079.
Grouping by greatest prime factor only gives A339195.
Row sums are 1 and A339360.
Cf. A001221, A005117, A005183, A014466, A019565, A187769, A209862.
Partition/composition orderings: A026791, A026792, A026793, A036036, A036037, A048793, A066099, A080577, A112798, A118457, A124734, A162247, A193073, A211992, A228100, A228531, A246688, A272020, A299755, A296774, A304038, A319247, A329631, A334301, A334302, A334439, A334442, A335122, A344086, A344087, A344088, A344089.
Partition/composition applications: A001793, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124.

nn=4;
GatherBy[SortBy[Select[Range[Times@@Prime/@Range[nn]],SquareFreeQ[#]&&PrimePi[FactorInteger[#][[-1,1]]]<=nn&],PrimeOmega],FactorInteger[#][[-1,1]]&]

A344090 Flattened tetrangle of strict integer partitions, sorted first by sum, then by length, then lexicographically.

Original entry on oeis.org

1, 2, 3, 2, 1, 4, 3, 1, 5, 3, 2, 4, 1, 6, 4, 2, 5, 1, 3, 2, 1, 7, 4, 3, 5, 2, 6, 1, 4, 2, 1, 8, 5, 3, 6, 2, 7, 1, 4, 3, 1, 5, 2, 1, 9, 5, 4, 6, 3, 7, 2, 8, 1, 4, 3, 2, 5, 3, 1, 6, 2, 1, 10, 6, 4, 7, 3, 8, 2, 9, 1, 5, 3, 2, 5, 4, 1, 6, 3, 1, 7, 2, 1, 4, 3, 2, 1

Offset: 0

Gus Wiseman, May 12 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (3)(21)
  4: (4)(31)
  5: (5)(32)(41)
  6: (6)(42)(51)(321)
  7: (7)(43)(52)(61)(421)
  8: (8)(53)(62)(71)(431)(521)
  9: (9)(54)(63)(72)(81)(432)(531)(621)

Wikiversity, Lexicographic and colexicographic order

Starting with reversed partitions gives A026793.
The version for compositions is A124734.
Showing partitions as Heinz numbers gives A246867.
The non-strict version is A334301 (reversed: A036036).
Ignoring length gives A344086 (reversed: A246688).
Same as A344089 with partitions reversed.
The version for revlex instead of lex is A344092.
A026791 reads off lexicographically ordered reversed partitions.
A080577 reads off reverse-lexicographically ordered partitions.
A112798 reads off reversed partitions by Heinz number.
A296150 reads off partitions by Heinz number.
Cf. A036037, A036043, A103921, A185974, A193073, A211992, A296774, A334302, A334433, A334435, A334438, A334439, A334440, A334441, A334442, A344091.

Table[Sort[Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,10}]

A098546 Table read by rows: row n has a term T(n,k) for each of the partition(n) partitions of n. T(n,k) = binomial(n,m) where m is the number of parts.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 6, 4, 1, 5, 10, 10, 10, 10, 5, 1, 6, 15, 15, 15, 20, 20, 20, 15, 15, 6, 1, 7, 21, 21, 21, 35, 35, 35, 35, 35, 35, 35, 21, 21, 7, 1, 8, 28, 28, 28, 28, 56, 56, 56, 56, 56, 70, 70, 70, 70, 70, 56, 56, 56, 28, 28, 8, 1, 9, 36, 36, 36, 36, 84, 84, 84, 84, 84, 84, 84

Offset: 1

Alford Arnold, Sep 14 2004

A035206 and A036038 were used to generate A049009 (Words over signatures). A098346 and A049019 provide another approach to the same end since A098346 times A049019 also yields A049009. (cf. A000312 and A000670).

Partitions are in Abramowitz and Stegun order. - Franklin T. Adams-Watters, Nov 20 2006

			A036042 begins 1 2 2 3 3 3 4 4 4 4 4 ...
A036043 begins 1 1 2 1 2 3 1 2 2 3 4 ...
so a(n) begins 1 2 1 3 3 1 4 6 6 4 1 ...
Table begins
.
1
2 1
3 3  1
4 6  6  4  1
5 10 10 10 10 5  1
6 15 15 20 15 20 15 20 15 6 1
.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A090657, A000041 (row lengths), A098545 (row sums), A036036, A036042, A036043.

Table[Sequence @@
  Map[Function[p, Binomial[n, Length[p]]], IntegerPartitions[n]], {n,
  1, 10}] (* Olivier Gérard, May 07 2024 *)

a(n) = Combin( A036042(n), A036043(n) )

A119441 Distribution of A063834 in Abramowitz and Stegun order.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 3, 4, 2, 1, 7, 5, 6, 3, 4, 2, 1, 11, 7, 10, 9, 5, 6, 8, 3, 4, 2, 1, 15, 11, 14, 15, 7, 10, 9, 12, 5, 6, 8, 3, 4, 2, 1, 22, 15, 22, 21, 25, 11, 14, 15, 20, 18, 7, 10, 9, 12, 16, 5, 6, 8, 3, 4, 2, 1, 30, 22, 30, 33, 35, 15, 22, 21

Offset: 1

Alford Arnold, May 19 2006

			1;
2, 1;
3, 2, 1;
5, 3, 4, 2, 1;
7, 5, 6, 3, 4, 2, 1;
T(5,2) = 5 because the second partition of 5 is 1+4 and 4 can be repartitioned in 5 different ways.
T(5,3) = 6 because the third partition of 5 is 2+3, where the 2 can be partitioned in 2 ways (2, 1+1) and the 3 can be partitioned in 3 ways (3, 1+2, 1+1+1), 6=2*3.
T(5,4) = 3 because the fourth partition of 5 is 1+1+3 and 3 can be partitioned in 3 different ways.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A063834, A119442, A000041 (row lengths and also first column)

# Compare two partitions (list) in AS order.
AScompare := proc(p1,p2)
    if nops(p1) > nops(p2) then
        return 1;
    elif nops(p1) < nops(p2) then
        return -1;
    else
        for i from 1 to nops(p1) do
            if op(i,p1) > op(i,p2) then
                return 1;
            elif op(i,p1) < op(i,p2) then
                return -1;
            end if;
        end do:
        return 0 ;
    end if;
end proc:
# Produce list of partitions in AS order
ASPrts := proc(n)
    local pi,insrt,p,ex ;
    pi := [] ;
    for p in combinat[partition](n) do
        insrt := 0 ;
        for ex from 1 to nops(pi) do
            if AScompare(p, op(ex,pi)) > 0 then
                insrt := ex ;
            end if;
        end do:
        if nops(pi) = 0 then
            pi := [p] ;
        elif insrt = 0 then
            pi := [p,op(pi)] ;
        elif insrt = nops(pi) then
            pi := [op(pi),p] ;
        else
            pi := [op(1..insrt,pi),p,op(insrt+1..nops(pi),pi)] ;
        end if;
    end do:
    return pi ;
end proc:
A119441 := proc(n,k)
    local pi,a,p ;
    pi := ASPrts(n)[k] ;
    a := 1 ;
    for p in pi do
        a := a*combinat[numbpart](p) ;
    end do:
    a ;
end proc:
for n from 1 to 10 do
    for k from 1 to A000041(n) do
        printf("%d,",A119441(n,k)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 12 2013

T(n,k) = product_{p=1..A036043(n,k)} A000041(c), 1<=k<=A000041(n), where c are the parts in the k-th partition of n. - R. J. Mathar, Jul 12 2013

A344087 Flattened tetrangle of strict integer partitions sorted first by sum, then colexicographically.

Original entry on oeis.org

1, 2, 2, 1, 3, 3, 1, 4, 4, 1, 3, 2, 5, 3, 2, 1, 5, 1, 4, 2, 6, 4, 2, 1, 6, 1, 5, 2, 4, 3, 7, 5, 2, 1, 4, 3, 1, 7, 1, 6, 2, 5, 3, 8, 6, 2, 1, 5, 3, 1, 8, 1, 4, 3, 2, 7, 2, 6, 3, 5, 4, 9, 4, 3, 2, 1, 7, 2, 1, 6, 3, 1, 5, 4, 1, 9, 1, 5, 3, 2, 8, 2, 7, 3, 6, 4, 10

Offset: 0

Gus Wiseman, May 11 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (21)(3)
  4: (31)(4)
  5: (41)(32)(5)
  6: (321)(51)(42)(6)
  7: (421)(61)(52)(43)(7)
  8: (521)(431)(71)(62)(53)(8)
  9: (621)(531)(81)(432)(72)(63)(54)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
Taking revlex instead of colex gives A118457.
The not necessarily strict version is A211992.
Taking lex instead of colex gives A344086.
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080576, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A246688, A272020, A299755, A296774, A304038, A319247, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344088, A344089, A344091.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Table[Sort[Select[IntegerPartitions[n],UnsameQ@@#&],colex],{n,0,10}]

cp's OEIS Frontend

A333485 Heinz numbers of all integer partitions sorted first by sum, then by decreasing length, and finally lexicographically. A code for the Fenner-Loizou tree A228100.

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A344089 Flattened tetrangle of reversed strict integer partitions, sorted first by length and then colexicographically.

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A111786 Array used to obtain the complete symmetric function in n variables in terms of the elementary symmetric functions; irregular triangle T(n,k), read by rows, with n >= 1 and 1 <= k <= A000041(n).

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A335123 Minimum part of the n-th integer partition in Abramowitz-Stegun order (sum/length/lex); a(0) = 0.

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A335124 Minimum part of the n-th reversed integer partition in Abramowitz-Stegun order; a(0) = 0.

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A344085 Triangle of squarefree numbers first grouped by greatest prime factor, then sorted by omega, then in increasing order, read by rows.

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A344090 Flattened tetrangle of strict integer partitions, sorted first by sum, then by length, then lexicographically.

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A098546 Table read by rows: row n has a term T(n,k) for each of the partition(n) partitions of n. T(n,k) = binomial(n,m) where m is the number of parts.

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